All Questions
Tagged with algebraic-combinatorics combinatorics
161
questions
3
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answers
42
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Exercise 2.2 In Stanley's Algebraic Combinatorics
This is Exercise $2.2$ In Stanley's Algebraic Combinatorics. I don't have much work to show because despite being stuck on this problem for a long time, I haven't got a clue how to start.
$\mathcal{C}...
- 3,100
0
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1
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36
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Exercise 9.6 in Algebraic Combinatorics by Stanley
Exercise 6 in chapter 9 of Algebraic Combinatorics by Stanley: Let $G$ be a finite graph on $p$ vertices with Laplacian matrix $L(G)$. Let $G'$ be obtained from $G$ by adding a new vertex $v$ and ...
4
votes
1
answer
57
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Exercise 8.6 of Algebraic Combinatorics by Stanley
Problem 6 in Chapter 8 of Algebraic Combinatorics by Stanley: Show that the total number of standard Young tableaux (SYT) with $n$ entries and at most two rows is ${n \choose \lfloor n/2 \rfloor}$. ...
1
vote
1
answer
38
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Exercise 7.2 in Algebraic Combinatorics by Stanley
This is exercise 2 in chapter 7 of Algebraic Combinatorics by Stanley.
For part (a), I first found the entire automorphism group. By labeling the root 1, and then numbering off the remaining vertices ...
1
vote
0
answers
37
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Exercise 3.1 of Algebraic Combinatorics by Richard Stanley
Exercise 3.1: Let $G$ be a (finite) graph with vertices $v_1, \ldots, v_p$. Assume that some power of the probability matrix $M(G)$ defined by $(3.1)$ has positive entries. (It's not hard to see that ...
0
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find the general solution the recurrence equation $b_n = 3b_{n-1} - b_{n-3}$
here are the steps I have done to try and find the general solution of this relation:
$$ b_n = 3b_{n-1} - b_{n-3}\\ = b^n = 3b^{n-1} - b^{n-3}$$
then divide by $b^{n-3}$ to get $$b^3 = 3b^2 - 1$$
then ...
- 15
1
vote
1
answer
44
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find the number of ways to distribute 30 students into 6 classes where there is max 6 students per classroom
here is the full question:
Use inclusion/exclusion to find the number of ways of distributing 30
students into six classrooms assuming that each classroom has a maximum capacity
of six students.
Let $...
- 15
1
vote
2
answers
81
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Identity of Schur polynomials
Let $p_n$ be the power sum symmetric polynomial,
$$p_n=x_1^n+x_2^n+\dots x_n^n$$
in $n$ variables, and let $s_\lambda$ be the Schur polynomials. I am new to Schur polynomials so I'm not sure what ...
- 83
0
votes
1
answer
36
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create a recurrence relation for the number of ways of creating an n-length sequence with a, b, and c where "cab" is only at the beginning
This is similar to a problem called forbidden sequence where you must find a recurrence relation for the number of ways of creating an n-length sequence using 0, 1, and 2 without the occurrence of the ...
- 15
0
votes
0
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14
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Isometric automorphisms of the ring of symmetric functions
I was trying to understand how special the $\omega$ involution on the ring of symmetric functions $\Lambda$ or $\Lambda^n$ (restriction to $n$ variables, just in case if by some magic, the situation ...
- 143
0
votes
0
answers
90
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Hyperplane Areangements and contraction.
I am trying to understand an idea presented in McNulty book, matriods a geometric introduction about the new hyperplane arrangement $\mathcal{A}^{''} = \{ H \cap H_x | H \in \mathcal{A}\}$ where $\...
- 3,139
1
vote
1
answer
144
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No minors isomorphic to $U_{0,1}$ and $U_{1,2}.$
I am trying to find all the matroids having no minors isomorphic to $U_{0,1}$ and $U_{1,2}.$
I know that the matroid $U_{0,n}$ is a matroid of rank zero with n edges and so it is just a vertex with n ...
- 95
3
votes
1
answer
101
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Understanding how to find the dual of a matroid.
I am trying to understand the following picture of a matroid and its dual but I found myself not understanding exactly what we are doing to find the dual:
]1
Roughly speaking, according to some ...
- 95
2
votes
1
answer
94
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Show that $C^*(e,B)$ is the unique cocircuit that is disjoint from $B - e.$
I want to prove the following question:
Let $B$ be a basis of a matroid $M.$ If $e \in B,$ denote $C_{M^*}(e, E(M) - B)$ by $C^*(e,B)$ and call it the fundamental cocircuit of $e$ with respect to $B.$\...
- 3,139
2
votes
1
answer
82
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Showing the isomorphism between (the geometric lattice)$\pi_n$ and the lattice $\mathcal{L}(M(K_n)).$
Here is the question I am trying to solve:
Show that the lattice of flats of $M(K_n)$ is isomorphic to the partition lattice $\pi_n$. Definition: $\pi_n$ is the set of partitions of $[n]$, partially ...
- 3,139